In This Lesson 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 What Is a DE? 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 Separable Equations First-Order Linear Exact Equations Applications What Is a Differential Equation? A DE is an equation involving a function and its derivatives . The order is the highest derivative present. A first-order DE has the form dy/dx = f(x, y).
General form: F(x, y, y') = 0
Solution: a function y = φ(x) satisfying the equation
General solution: family of solutions with arbitrary constant C
Particular solution: satisfies an initial condition y(x₀) = y₀
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Form: dy/dx = g(x)·h(y)
Method: (1/h(y)) dy = g(x) dx → integrate both sides
Example: dy/dx = xy (1/y) dy = x dx → ln|y| = x²/2 + C → y = Ae^(x²/2)
Uses integration and exponential functions .
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Form: dy/dx + P(x)y = Q(x)
Integrating factor: μ(x) = e^(∫P(x)dx)
Solution: y = (1/μ) ∫ μ·Q dx
The integrating factor technique converts a non-separable DE into an exact derivative. This method uses the product rule in reverse and relies on integration techniques .
Exact Equations
Form: M(x,y)dx + N(x,y)dy = 0
Exact if: ∂M/∂y = ∂N/∂x
Solution: find F where ∂F/∂x = M and ∂F/∂y = N; then F(x,y) = C
Exactness connects to partial derivatives and conservative vector fields. Non-exact equations can sometimes be made exact with an integrating factor.
Applications 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 Exponential growth/decay: dy/dt = ky → y = y₀eᵏᵗ (population, radioactive decay) Newton's cooling: dT/dt = −k(T − Tₐ) → T = Tₐ + (T₀ − Tₐ)e⁻ᵏᵗ Mixing problems: Rate in − rate out → first-order linear DE Logistic growth: dP/dt = rP(1 − P/K) → S-shaped curve — uses partial fractions to integrate
First-order DEs appear in every scientific field. In
systems of DEs , x' = Ax, eigenvalues determine whether solutions grow, decay, or oscillate. In
probability , the exponential distribution's memoryless property comes from the DE: f'(t) = −λf(t).
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